# Thread: State the domain where the function is analytic

1. ## State the domain where the function is analytic

State the domain where the function is analytic.
$f(z) = \frac{1}{1-z^4}$

From what I have studied, f(z) is analytic on a domain D, if f(z) is differentiable at all points in D.

However, I don't know where I should begin. And I think that $f(z)$ is not analytic at $1-z^4=0$.

Anyway, how to find the domain, how to show in details to solve this.

Thank you.

2. Originally Posted by noppawit
State the domain where the function is analytic.
$f(z) = \frac{1}{1-z^4}$

From what I have studied, f(z) is analytic on a domain D, if f(z) is differentiable at all points in D.

However, I don't know where I should begin. And I think that $f(z)$ is not analytic at $1-z^4=0$.

Anyway, how to find the domain, how to show in details to solve this.

Thank you.
As you have said, consider when $1-z^4=0$.

This would construct the 4th roots of unity $z=i,\,z=-i,\,z=1,\,z=-1$: $f\!\left(z\right)$ is not differentiable at these values.

Thus, our domain would be $D=\mathbb{C}\backslash\{1,i,-1,-i\}$.